Originally posted by SwissGambitI really appreciate the effort, guys. I have some practicing to do to master this technique. Thanks again.
I already had found that easier path. See post 4 on page 1:(x-273.15)*1.8+32=-xGo back to:
1.8x -491.67+32=-x
-459.67=-2.8x
x=164.1678571-459.67=-2.8xMultiply both sides by 100:-45967=-280xBut I was showing sonhouse a general method for converting ANY number with a repeating decimal to a fraction. I just happened to use that number as an example.
x=45967/280
5 edits
Originally posted by sonhouseWhen you see something in an equation in the form of a decimal just convert it to it's fractional equivalent (do this for all terms). Then just simplify and solve.
Well my calculator says 45967/280 covers the 164.1678571 part exactly but it misses out on the actual repeaters, the 428571's that repeat forever.
How did you suss out that fraction anyway? You have fraction sniffing software?
i.e. (K-273.15)*1.8 +32 = +/-K becomes
((100K - 27315)/100)(9/5) + 32 = +/-K
==> 9(20K - 5463) +3200 = +/-100K
==> 180 +/- 100K = 49167 - 3200
==> K = 45967/(180 +/- 100)
Also, when you see a recuring decimal, you can quickly convert to its exact fractional form, by expressing it as a sum of the repeating and non repeating parts (where the repeating part can be written as x/D(x) where x is the repeated term, and D(x) is 10^(number of digits in x) -1. i.e. with x = 123 then 0.123123123123 = 123/(D(123)) = 123/999
This works because if (relabeling x)
[1] x = f + bar(y) (f is some fraction, y a repeated term in the decimal part) then
(10^d)x = (10^d)f + y + bar(y) (d is the number of digits in y)
==> (10^d - 1)x = (10^d - 1)f + y (subtracting [1])
==> x = f +y/(10^d - 1)
Originally posted by AgergUgh, all that extra clutter in the intermediate steps.
When you see something in an equation in the form of a decimal just convert it to it's fractional equivalent (do this for all terms). Then just simplify and solve.
i.e. (K-273.15)*1.8 +32 = +/-K becomes
((100K - 27315)/100)(9/5) + 32 = +/-K
==> 9(20K - 5463) +3200 = +/-100K
==> 180 +/- 100K = 49167 - 3200
==> K = 45967/(180 +/- 100)
(x-273.15)*1.8+32=+/-x
1.8x-491.67+32=+/-x
-459.67=-1.8+/-1(x)
And the fraction emerges naturally:
x = 45967/(180 +/- 100)
Originally posted by SwissGambitOk...I think it's a matter of preference though - I prefer to clear out decimals as early as possible.
Ugh, all that extra clutter in the intermediate steps.
(x-273.15)*1.8+32=+/-x
1.8x-491.67+32=+/-x
-459.67=-1.8+/-1(x)
And the fraction emerges naturally:
x = 45967/(180 +/- 100)
Originally posted by AgergHow does this method handle a number like 0.000 123 123 123 123 123... ?
Also, when you see a recuring decimal, you can quickly convert to its exact fractional form, by expressing it as a sum of the repeating and non repeating parts (where the repeating part can be written as x/D(x) where x is the repeated term, and D(x) is 10^(number of digits in x) -1. i.e. with x = 123 then 0.123123123123 = 123/(D(123)) = 123/999
This works bec ...[text shortened]... f digits in y)
==> (10^d - 1)x = (10^d - 1)f + y (subtracting [1])
==> x = f +y/(10^d - 1)
- Joined
- 26 Apr 03
- Moves
- 26771
2 edits
Originally posted by SwissGambitjust multiply by enough "0s" to left shift by the length of 1 repeating part:
How does this method handle a number like 0.000 123 123 123 123 123... ?
x = 0.000 123 123 123 123 123...
1000x = 0.123 123 123 123...
999x = 0.123
999x = 123/1000
x = 123/999000
x = (3*41)/(3*333000)
x = 41/333000
If you like recurring decimals, continued fractions are really good too, they turn any root of any number into a recurring fraction.
8 edits
Originally posted by SwissGambithmm...good point, I retract the method and "proof" - if there is non-recurring decimal part, account of this (in particular the length) needs to be made when looking the recurring part (as iamatiger suggests)
How does this method handle a number like 0.000 123 123 123 123 123... ?
Damn that's embarrassing! 😞
Originally posted by AgergNah, no need to retract - just modify the method as iamatiger did, and it's fine.
hmm...good point, I retract the method and "proof" - if there is non-recurring decimal part, account of this (in particular the length) needs to be made when looking the recurring part (as iamatiger suggests)
Damn that's embarrassing! 😞